Rational self-maps of projective surfaces with a regular iterate

The paper proves the trichotomy and the uniform bound i ∈ {1,2,3,4,6,8,12} for eventually regular, non-invertible self-maps on smooth projective surfaces (Theorems 1.1 and 1.4) via a robust strategy: existence of totally invariant loci (Proposition 1.12), an étaleness argument when κ(X) ≥ 0, an Amerik-type structural result for P^1-bundles, and a careful toric reduction combining the P^2 and Hirzebruch cases (Sections 8–10). These arguments are coherent and cross-referenced inside the paper, e.g., Theorem 1.4 and its proof outline and completion , Proposition 1.12 , the P^1-bundle factorization and regularity (Proposition 9.1, Theorem 9.2) , and the toric case (Theorem 8.9, Corollary 8.8, and the P^2 case Theorem 8.1) . By contrast, the candidate solution contains substantive gaps: (i) in the κ(X) ≥ 0 case, the intersection argument on the graph neglects that the ramification divisor R can intersect a σ-exceptional curve negatively; (ii) in the P^1-bundle case, the “poles cannot cancel under composition” claim is unjustified; and (iii) in the toric case, it incorrectly assumes a general morphism or rational self-map induces an action on the fan rays, which holds only for torus-equivariant (toric) morphisms. Hence the model’s reasoning is unsound even though its final conclusions match the paper.